Latex/Lyx: not just retyping from paper, but live problem-solving?

Has anyone tried to use software not just for typing math to produce good-looking documents after solving problems/doing research with paper&pencil, but for actually doing research/problem solving with lots of maths directly in electronic form? Has anyone managed to have the same productivity as with paper and pencil?

I have been trying to use LaTeX this way for some time, and I found that my productivity is far lower than with paper&pencil, in particular because LaTeX is non-WYSIWYG and you have to compile to check results. Caring about that takes part of my attention away from the actual problem solving. I currently have somewhat more success with LyX, which is WYSIWYG, however I still think that my productivity is lower.

Don't do that then. TeX and LaTeX are intended to be WYSIWYM (what you say is what you mean) rather than WYSIWYG. Write what you mean, then fix it later so it looks right.

With regard to your question, our group doesn't just use LaTeX for math on a forum. I work on a rather large (> 500 KSLOC) highly mathematical software package whose main body of documentation is entirely in LaTeX. The math is but a small part of that documentation. There's also requirements (for which we have formal LaTeX macros), formal and informal descriptions of the software, test results, traceability (automatically generated!), and metrics (also automatically generated). Other groups have used our documentation style and our LaTeX macros, so it now is used on well over a million lines of code, much of which is mathematical programming.

I will do it only when I am trying to solve a problem that I am confident is easy to solve, and does not have large expressions. I don't even bother trying to compile until the end, I just read the previous tex code to see what I was writing out (hence the no long expression rule!), and will occasionally pull out a pencil and paper to verify a calculation that is more tedious.

It is certainly the case that writing it up in tex as a way of solving the problem is not the intended use of the language, so I wouldn't bother trying it too much if you find it difficult.

What I often do is a sketch using paper & pencil but then I switch to latex to get all the signs and factors right, leaving only the important steps. The chances to find and understand it some months later are far higher.